... I was interested in the cases when intersection of ANY arrangement of Schubert cells has the M-property. You are interested in the cases when there EXISTS some open set of arrangements of real Schubert cells which intersects in all real points. I want to adjust my conjecture to your case.
Let S1,..., Sk be some arrangement of real Schubert cells such that S1\cap...\cap Sk is 0-dimensional and consists of the necessary number of points (counted with multiplicities).
Conjecture. Intersection of any SUBSET of the above cells has the M-property, i.e. the sum of Betti numbers is the same as for the complexified arrangement.
If your question is open for arrangements of Schubert cells in the space of
complete flags let me know. I may have a wild guess how one constructs such
arrangements (but this is only a wild guess at the moment).
B.Sh.